Overdetermined elliptic problems in nontrivial contractible domains of the sphere

IF 2.1 1区 数学 Q1 MATHEMATICS
David Ruiz , Pieralberto Sicbaldi , Jing Wu
{"title":"Overdetermined elliptic problems in nontrivial contractible domains of the sphere","authors":"David Ruiz ,&nbsp;Pieralberto Sicbaldi ,&nbsp;Jing Wu","doi":"10.1016/j.matpur.2023.10.009","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we prove the existence of nontrivial contractible domains <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>, such that the overdetermined elliptic problem<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mo>−</mo><mi>ε</mi><msub><mrow><mi>Δ</mi></mrow><mrow><mi>g</mi></mrow></msub><mi>u</mi><mo>+</mo><mi>u</mi><mo>−</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><mn>0</mn></mtd><mtd><mtext>in Ω, </mtext></mtd></mtr><mtr><mtd><mi>u</mi><mo>&gt;</mo><mn>0</mn></mtd><mtd><mtext>in Ω, </mtext></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mn>0</mn></mtd><mtd><mtext>on ∂Ω, </mtext></mtd></mtr><mtr><mtd><msub><mrow><mo>∂</mo></mrow><mrow><mi>ν</mi></mrow></msub><mi>u</mi><mo>=</mo><mtext>constant</mtext></mtd><mtd><mtext>on ∂Ω, </mtext></mtd></mtr></mtable></mrow></math></span></span></span> admits a positive solution. Here <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span> is the Laplace-Beltrami operator in the unit sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> with respect to the canonical round metric <em>g</em>, <span><math><mi>ε</mi><mo>&gt;</mo><mn>0</mn></math></span> is a small real parameter and <span><math><mn>1</mn><mo>&lt;</mo><mi>p</mi><mo>&lt;</mo><mfrac><mrow><mi>d</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>d</mi><mo>−</mo><mn>2</mn></mrow></mfrac></math></span> (<span><math><mi>p</mi><mo>&gt;</mo><mn>1</mn></math></span> if <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>). These domains are perturbations of <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>∖</mo><mi>D</mi></math></span>, where <em>D</em><span><span> is a small geodesic ball. This shows in particular that Serrin's theorem for </span>overdetermined problems<span> in the Euclidean space cannot be generalized to the sphere even for contractible domains.</span></span></p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"180 ","pages":"Pages 151-187"},"PeriodicalIF":2.1000,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal de Mathematiques Pures et Appliquees","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021782423001502","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1

Abstract

In this paper, we prove the existence of nontrivial contractible domains ΩSd, d2, such that the overdetermined elliptic problem{εΔgu+uup=0in Ω, u>0in Ω, u=0on ∂Ω, νu=constanton ∂Ω,  admits a positive solution. Here Δg is the Laplace-Beltrami operator in the unit sphere Sd with respect to the canonical round metric g, ε>0 is a small real parameter and 1<p<d+2d2 (p>1 if d=2). These domains are perturbations of SdD, where D is a small geodesic ball. This shows in particular that Serrin's theorem for overdetermined problems in the Euclidean space cannot be generalized to the sphere even for contractible domains.

球面非平凡可缩区域上的超定椭圆问题
在本文中,我们证明了非平凡可缩域Ω∧Sd, d≥2的存在性,使得超定椭圆问题{−εΔgu+u−up=0in Ω, u>0in Ω, u=0on∂Ω,∂νu=constanton∂Ω有一个正解。这里Δg是单位球Sd中关于正则圆度规g的拉普拉斯-贝尔特拉米算子,ε>0是一个小实参数,1<p<d+2d - 2(如果d=2, p>1)。这些区域是Sd²D的摄动,其中D是一个小的测地线球。这特别说明了欧氏空间中关于超定问题的Serrin定理不能推广到球面上,即使在可缩域上也是如此。
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来源期刊
CiteScore
4.30
自引率
0.00%
发文量
84
审稿时长
6 months
期刊介绍: Published from 1836 by the leading French mathematicians, the Journal des Mathématiques Pures et Appliquées is the second oldest international mathematical journal in the world. It was founded by Joseph Liouville and published continuously by leading French Mathematicians - among the latest: Jean Leray, Jacques-Louis Lions, Paul Malliavin and presently Pierre-Louis Lions.
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